Great circles
See also - Great circle calculator
What is a great circle?
Look at the map below, in particular the lines joining London (United Kingdom) and Wellington (New Zealand)...
Look at the map below, in particular the lines joining London (United Kingdom) and Wellington (New Zealand)...
Although the red line might look like the shortest route connecting the two cities on the map, the yellow line is in fact the shortest path. This yellow line is called a great circle.
Why don't the red and yellow lines take the same route?
Because the Earth is (roughly) a sphere in shape, and the map above is a flat surface. Therefore a line drawn on a sphere will become distorted when it is shown on the map.
So the Earth isn't a sphere, either?
Not quite, no. The Earth actually resembles a sphere that has been squashed slightly at the poles and bulges a little at the equator. However, unless high accuracy is needed, it is sufficient (and much easier) to assume the Earth is spherical.
How can I calculate the length of a great circle?
Assuming the Earth is a sphere then this is relatively simple.
To start with, you'll need the latitude and longitude of the two points on the Earth in decimal degree form. For example,
If we call the latitudes of the start and finish point φ1 and φ2 respectively, and similarly label the two longitudes as L1 and L2, we then find d using the formula...
d = acos[ sin(φ1) sin(φ2) + cos(φ1) cos(φ2) cos(L1 - L2) ] (sines/cosines in degrees)
To convert d into miles, kilometres, or some other unit, you need to multiply d by π, divide by 180, and then multiply by the radius of the Earth in the chosen unit.
For example, using London and Wellington as our points, we find d to be 169.221451. To get this in kilometres, we multiply 169.221451 by π (= 3.141593) to get 531.624869, divide by 180 to get 2.95347150, and then multiply by the radius of the Earth in kilometres (= 6378.14) to get the final answer of 18837.7 kilometres.
How can I calculate the distance through the Earth between two points?
Assuming you have calculated the length of the great circle already (see above), this is very easy. If x is your chosen unit of measurement (e.g. miles, kilometres) and r is the radius of the Earth in the chosen unit of measurement, the distance through the Earth, y, is given by...
y = 2r sin( 90g / πr ) (sine in degrees)
y = 2r sin( g / 2r ) (sine in radians)
How can I plot the path of a great circle?
This is a little more tricky. Below I explain how my great circle calculator program does this, but there are probably better methods out there.To start, you will need to have figures for φ1, L1, φ2, L2 and d (see above). Then calculate angle α by means of the formula...
α = acos[ { sin(φ2) - cos(d) sin(φ1) } / { sin(d) cos(φ1) } ]
Now choose a number f between 0 and 1 which specifies how far along the great circle you are. For example, to find the latitude and longitude of the point exactly halfway along the great circle, set f to 0.5.Now calculate
g = df
The latitude of the required point, φ, is then given by...
φ = asin[ sin(φ1) cos(g) + cos(φ1) sin(g) cos(α) ]
To calculate the longitude of the required point, L, calculate the quantities p and ΔL by the formulae
p = sin(L1 - L2)
ΔL = acos[ { cos(g) - sin(φ) sin(φ1) } / { cos(φ) cos(φ1) } ]
There are then three cases:
If ΔL is undefined, then L = L2
If ΔL is defined and p ≥ 0 then L = L1 + ΔL
If ΔL is defined and p < 0 then L = L1 - ΔL
If L falls outside the range -180 to 180, simply add or subtract multiples of 360 until it falls within this range.Hence by varying f from 0 to 1, it is possible to plot out the path of the great circle.
How can I calculate the length of a great circle without assuming the Earth is a sphere?
You can use the formula due to H Andoyer. I first came across the algorithm in Jean Meeus' Astronomical Algorithms, but the interested reader can also find it online at the Museum of HP calculators website (direct link; the formulae of interest are in section 1a and 1b).
And what if I don't want to do all the calculations?
Simple - use my great circle calculator program!
Why don't the red and yellow lines take the same route?
Because the Earth is (roughly) a sphere in shape, and the map above is a flat surface. Therefore a line drawn on a sphere will become distorted when it is shown on the map.
So the Earth isn't a sphere, either?
Not quite, no. The Earth actually resembles a sphere that has been squashed slightly at the poles and bulges a little at the equator. However, unless high accuracy is needed, it is sufficient (and much easier) to assume the Earth is spherical.
How can I calculate the length of a great circle?
Assuming the Earth is a sphere then this is relatively simple.
To start with, you'll need the latitude and longitude of the two points on the Earth in decimal degree form. For example,
- London has latitude 51° 31' N and longitude 000° 08' W, which in decimal degrees, is latitude 51.516667 and longitude -0.133333.
- Wellington has latitude 41° 19' S and longitude 174° 47' E, which is equivalent to latitude -41.316666 and longitude 174.783333.
If we call the latitudes of the start and finish point φ1 and φ2 respectively, and similarly label the two longitudes as L1 and L2, we then find d using the formula...
d = acos[ sin(φ1) sin(φ2) + cos(φ1) cos(φ2) cos(L1 - L2) ] (sines/cosines in degrees)
To convert d into miles, kilometres, or some other unit, you need to multiply d by π, divide by 180, and then multiply by the radius of the Earth in the chosen unit.
For example, using London and Wellington as our points, we find d to be 169.221451. To get this in kilometres, we multiply 169.221451 by π (= 3.141593) to get 531.624869, divide by 180 to get 2.95347150, and then multiply by the radius of the Earth in kilometres (= 6378.14) to get the final answer of 18837.7 kilometres.
How can I calculate the distance through the Earth between two points?
Assuming you have calculated the length of the great circle already (see above), this is very easy. If x is your chosen unit of measurement (e.g. miles, kilometres) and r is the radius of the Earth in the chosen unit of measurement, the distance through the Earth, y, is given by...
y = 2r sin( 90g / πr ) (sine in degrees)
y = 2r sin( g / 2r ) (sine in radians)
How can I plot the path of a great circle?
This is a little more tricky. Below I explain how my great circle calculator program does this, but there are probably better methods out there.To start, you will need to have figures for φ1, L1, φ2, L2 and d (see above). Then calculate angle α by means of the formula...
α = acos[ { sin(φ2) - cos(d) sin(φ1) } / { sin(d) cos(φ1) } ]
Now choose a number f between 0 and 1 which specifies how far along the great circle you are. For example, to find the latitude and longitude of the point exactly halfway along the great circle, set f to 0.5.Now calculate
g = df
The latitude of the required point, φ, is then given by...
φ = asin[ sin(φ1) cos(g) + cos(φ1) sin(g) cos(α) ]
To calculate the longitude of the required point, L, calculate the quantities p and ΔL by the formulae
p = sin(L1 - L2)
ΔL = acos[ { cos(g) - sin(φ) sin(φ1) } / { cos(φ) cos(φ1) } ]
There are then three cases:
If ΔL is undefined, then L = L2
If ΔL is defined and p ≥ 0 then L = L1 + ΔL
If ΔL is defined and p < 0 then L = L1 - ΔL
If L falls outside the range -180 to 180, simply add or subtract multiples of 360 until it falls within this range.Hence by varying f from 0 to 1, it is possible to plot out the path of the great circle.
How can I calculate the length of a great circle without assuming the Earth is a sphere?
You can use the formula due to H Andoyer. I first came across the algorithm in Jean Meeus' Astronomical Algorithms, but the interested reader can also find it online at the Museum of HP calculators website (direct link; the formulae of interest are in section 1a and 1b).
And what if I don't want to do all the calculations?
Simple - use my great circle calculator program!
